Integral coefficient rings and homological dimensions of algebras

@misc{liu2026integral,
  title = {{Integral coefficient rings and homological dimensions of algebras}},
  author = {Yu-Zhe Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.00143},
  eprint = {2609.00143},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00143},
}

TY  - UNPB
TI  - Integral coefficient rings and homological dimensions of algebras
AU  - Yu-Zhe Liu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00143
UR  - https://arxiv.org/abs/2609.00143
AB  - We define the integral profiles of all modules and introduce integral coefficient rings ^𝒫mkern−6.5mu&mkern−5.5mu_I(A) for all finite-dimensional complex algebras A. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules M≅ N are isomorphic if and only if their integral profiles are similar, i.e., M≅ N if and only if ∫ M ∼ ∫ N. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of A (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of A is finite if and only if the embedded integral profile of top(A) lies in ^𝒫mkern−6.5mu&mkern−5.5mu_I(A)[x]; (6) the finitistic dimension of A is finite if and only if each embedded integral profile of M lying in ^𝒫mkern−6.5mu&mkern−5.5mu_I(A)[x] implies that its degree is less than or equal to a fixing integer d∈ℕ^+; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra A satisfies one of them, then its finitistic dimension is finite.
ER  -